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Extensions of inequalities for sector matrices
Journal of Inequalities and Applications volume 2019, Article number: 118 (2019)
Abstract
In this note, we first prove an inequality for sector matrices. This complements a result due to Kittaneh and Sakkijha (Linear Multilinear Algebra, 2018, https://doi.org/10.1080/03081087.2018.1441800) concerning accretive–dissipative matrices. And then we present two singular value inequalities for sector matrices which are similar to Yang and Lu’s inequalities (J. Inequal. Appl. 2018:183, 2018).
1 Introduction
We denote by \(\mathbb{M}_{n}(\mathbb{C})\) the set of \(n\times n\) complex matrices. For \(A\in \mathbb{M}_{n}(\mathbb{C})\), the conjugate transpose of A is denoted by \(A^{*}\), and the matrices \(\mathfrak{R}A= \frac{1}{2}(A+A^{*})\) and \(\mathfrak{I}A=\frac{1}{2i}(A-A^{*})\) are called the real part and imaginary part of A, respectively (e.g., [1, p. 6] and [6, p. 7]). Recall that a norm \(\|\cdot \|\) on \(\mathbb{M}_{n}(\mathbb{C})\) is unitarily invariant if \(\|UAV\|=\|A\|\) for any \(A\in \mathbb{M}_{n}(\mathbb{C})\) and unitarily matrices \(U, V \in \mathbb{M}_{n}(\mathbb{C})\). For \(p \geq 1\), the Schatten p-norm of \(A\in \mathbb{M}_{n}(\mathbb{C})\) is defined as \(\|A\|_{p}= (\sum_{j=1}^{n}\sigma _{j}^{p}(A) )^{\frac{1}{p}}\). If the eigenvalues of a square matrix \(A\in \mathbb{M}_{n}(\mathbb{C})\) are all real, then we denote \(\lambda _{j}(A)\) the jth largest eigenvalue of A. The singular values of a complex matrix \(A\in \mathbb{M}_{n}\) are the eigenvalues of \(|A|:=(A^{*}A)^{\frac{1}{2}}\), and we denote \(\sigma _{j}(A):=\lambda _{j}(|A|)\) the jth largest singular value of A. A positive semidefinite matrix A will be expressed as \(A\geq 0\). Likewise, we write \(A>0\) to refer that A is a positive definite matrix.
\(A\in \mathbb{M}_{n}(\mathbb{C})\) is an accretive-dissipative matrix if \(\mathfrak{R} A\) and \(\mathfrak{I} A\) are both positive semidefinite. This class of matrices has recently been considered by Lin [9] and Lin and Zhou [11].
Let \(H\in \mathbb{M}_{n}(\mathbb{C})\) be a Hermitian matrix and let f be a real-valued function defined on an interval containing all the eigenvalues of H. Then \(f(H)\) is well defined through spectral decomposition. f is called matrix concave if \(f(\alpha A+(1-\alpha )B)\geq \alpha f(A)+(1-\alpha )f(B)\) for any two Hermitian matrices \(A,B\in \mathbb{M}_{n}(\mathbb{C})\) and all \(\alpha \in [0,1]\).
The numerical range of \(A\in \mathbb{M}_{n}(\mathbb{C})\) is defined by
For \(\alpha \in [0, \pi /2)\), let
be a sector region on the complex plane. A matrix whose numerical range is contained in a sector region \(S_{\alpha }\) is called a sector matrix [10]. Recent research interest in this class of matrices starts with a resolution of a problem from numerical analysis [3]. Some research results on sector matrices can be found in [3, 7, 8, 10, 13].
Kittaneh and Sakkijha [7] proved the following Schatten-p norm inequalities.
Theorem 1.1
(see [7, Theorem 2.7])
Let \(S, T \in \mathbb{M}_{n}(\mathbb{C})\) be accretive-dissipative. Then
Recently, Yang and Lu [12] gave a generalization of Theorem 1.1.
Theorem 1.2
(see [12, Theorem 2.3])
Let \(A_{1},\ldots, A_{n} \in \mathbb{M}_{n}(\mathbb{C})\) be accretive-dissipative. Then
In [5], Garg and Aujla presented the following inequalities:
where \(A, B \in \mathbb{M}_{n}(\mathbb{C})\) and \(f: [0,\infty )\rightarrow [0,\infty )\) is a matrix concave function.
If \(A, B \geq 0, r=1\) and \(f(X)=X\) for any \(X \in \mathbb{M}_{n}( \mathbb{C})\) in (1) and (2), then
and
Based on the above inequalities, Yang and Lu (see [12, Theorem 2.7]) gave the inequalities for sector matrices which removed the absolute values in (1) and (2) from the right sides as follows.
Theorem 1.3
(see [12, Theorem 2.7])
Let \(A, B\in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A), W(B)\subseteq S_{\alpha }\). Then, for \(1 \leq k \leq n\),
and
By Fan’s dominance principle [1, p. 93], the theorem below follows from (4) and (5).
Theorem 1.4
(see [12, Corollary 2.8])
Let \(A, B\in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A), W(B) \subseteq S_{\alpha }\). Then
and
In this paper, we will extend Theorem 1.2 to sector matrices. Furthermore, we present some inequalities for sector matrices which are similar to the inequalities (4) and (5). However, in some cases, our results are stronger than (4) and (5), respectively.
2 Main results
We begin this section with some lemmas which are useful to establish our main results.
Lemma 2.1
(see [1, p. 73])
Let \(A \in \mathbb{M}_{n}(\mathbb{C})\). Then
Consequently,
Lemma 2.2
(see [13, Lemma 3.1])
Let \(A \in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then
Lemma 2.3
(see [2, (4)])
Let \(A_{1},\ldots,A_{n} \in \mathbb{M} _{n}(\mathbb{C})\) be positive semidefinite. Then, for \(p \geq 1\),
Lemma 2.4
(see [4, Theorem 4.1])
Let \(A \in \mathbb{M}_{n}( \mathbb{C})\) be such that \(W(A)\subseteq S_{\alpha }\). Then
The above inequality implies that there exists a unitary matrix \(U\in \mathbb{M}_{n}(\mathbb{C})\) such that
Lemma 2.5
(see [1, Theorem III.5.6])
Let \(A,B \in \mathbb{M} _{n}(\mathbb{C})\). Then there exist unitary matrices \(U,V \in \mathbb{M}_{n}(\mathbb{C})\) such that
For the above preparation, we present the first main result which is an extension of Theorem 1.1.
Theorem 2.6
Let \(A_{1}, \ldots, A_{n}\in \mathbb{M}_{n}(\mathbb{C})\) be sector matrices. Then
Proof
Let \(A_{j}=B_{j}+iC_{j}\) be the Cartesian decompositions of \(A_{j}\), \(j=1, \ldots,n\). Then we have
which proves the first inequality.
To prove the second inequality, compute
which completes the proof. □
Remark 2.7
Theorem 1.2 is immediate by setting \(\alpha =\frac{\pi }{4}\) in (14). Moreover, when \(n=2\) and \(\alpha =\frac{\pi }{4}\) in Theorem 2.6, our result is Theorem 1.1.
Next, we end this section with a generalization of singular value inequality for two positive semidefinite matrices to sector matrices.
Theorem 2.8
Let \(A, B\in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A), W(B) \subseteq S_{\alpha }\). Then, for \(1 \leq k\leq n\), the following assertions hold:
and
Proof
Compute
where \(U,V\) are unitary matrices.
To prove (16), compute
where \(U_{j}\) and \(V_{j}\), \(j=1,2,3\), are unitary matrices.
Thus
This completes the proof. □
The following examples show that neither (4) nor (15) is uniformly better than the other.
Example 2.9
Let
and
be such that \(W(A), W(B)\subseteq S_{\frac{\pi }{4}}\).
For the right side of the inequality (4),
When \(k=2\), for the right side of the inequality (15),
This shows that (15) is stronger than (4).
Example 2.10
If
and
are such that \(W(A), W(B)\subseteq S_{\frac{\pi }{4}}\), we also suppose \(k=2\).
For the right side of the inequality (4),
For the right side of the inequality (15),
The example implies that the bound in (15) is weaker than that in (4).
Remark 2.11
Actually, the above examples also show that the inequalities (5) and (16) are not comparable.
Corollary 2.12
Let \(A, B\in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A), W(B) \subseteq S_{\alpha }\). Then, for all unitarily invariant norms \(\|\cdot \|\) on \(\mathbb{M}_{n}(\mathbb{C})\),
and
Proof
From (15) and (16), we obtain for \(1 \leq k\leq n\)
and
By the property that weak log-majorization implies weak majorization, we have for \(1 \leq k\leq n\)
and
Then, by the Cauchy–Schwarz inequality, for \(1 \leq k\leq n\)
and
By Fan’s dominance principle [1, p. 93], we have
and
Let \(A+B=U|A+B|, I_{n}+A+B=V|I_{n}+A+B|\) be the polar decomposition of \(A+B\) and \(I_{n}+A+B\), respectively, where U and V are unitary matrices. Thus, by (19), we have
Similarly, by (20) we have
which completes the proof. □
Remark 2.13
By computing Examples 2.9 and 2.10, it should be noticed here that neither (6) nor (17) is uniformly better than the other. When comparing the inequality (7) with (18), the same conclusion can be drawn.
Taking \(k=n\) in Theorem 2.8, we get the following corollary.
Corollary 2.14
Let \(A, B\in \mathbb{M}_{n}(\mathbb{C})\) be such that \(W(A), W(B) \subseteq S_{\alpha }\). Then
and
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This paper is supported by Foundation for Scientific Research by the Ministry of Education of Hainan, China (No. Hnky2019ZD-13) and Excellent Young Teachers Growth Program of Hainan Normal University, Haikou, China.
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Lin, S., Fu, X. Extensions of inequalities for sector matrices. J Inequal Appl 2019, 118 (2019). https://doi.org/10.1186/s13660-019-2069-8
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DOI: https://doi.org/10.1186/s13660-019-2069-8